2007/02/21 by John A. Baldwin, Baldwin, John A.
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG) #math.GT #math.SG
paper · pdf · doi:10.48550/arxiv.math/0702603
15 pages, 4 figures. This updated version contains a generalization of the main theorem from before, with consequences relating to the planarity of open books, and Heegaard Floer L-spaces
arxiv created 2007/04/10 · arxiv updated 2009/12/01
Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Yg, Yh, and Yhg be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural under a comultiplication map on Heegaard Floer homology. It follows that if the contact invariants associated to the open books (S, g) and (S, h) are non-zero then the contact invariant associated to the open book (S, hg) is also non-zero. We extend this comultiplication to a map on HF+, and as a result we obtain obstructions to the three-manifold Yhg being an L-space. We also use this to find restrictions on contact structures which are compatible with planar open books.